How to prove that an exterior angle of a cyclic quadrilateral is equal to its opposite interior angle?

Question

5902 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-26T12:34:04

Use two other theorems to show that this is true.

This uses two other theorems.

#1.# The opposite angles of a cyclic quad are supplementary.

So #x + y =180°#

#2.# Adjacent angles on a straight line are supplementary.

So #x_1 +x_2 = 180°#

An exterior angle of any shape is formed by extending a side.

Therefore for cyclic quadrilateral with a vertex #X# lengthened to create an exterior angle and an opposite vertex #Y# ,

#x_1 + x_2 = 180° " "# (adj angles on str line)
#x_1 +y = 180°" "#(opp angles cyclic quad)

#:. x_2 = y#

The exterior angle of a cyclic quad is equal to the interior opposite angle.